I am not sure what kind of dryer I have, but it squeaks the whole time it's drying. Another major problem with this dryer is that if we put more than 5 towels in it, it won't even turn! I am not sure what is wrong with it. Any advice?

The belt or something is off balance. That might have happened if you dry tennis shoes in it,
My kids have messed up 2 of my dryers from doing that. You have to get someone to take the back off and check if it is the drum and/or belt and have it fixed. Or buy another dryer.
I got one of those mesh cotton bags and have them put their tennis shoes in it and hang it inside the door so that never happened again.

I have never put shoes in there. Do you think that it would be cheaper to fix it, or just get a new one? I am sure if we get the parts, we can fix it ourselves maybe. It's 5 years old.

Cheaper to fix. Belt is probably streched and needs replacing. The noise can also be the idler pulley and/or the rollers. Not hard to check and repair. Inexpensive